### Critical case of nonlinear Schrödinger equations with inverse-square potentials on bounded domains

Nonlinear Schrödinger equations (NLS)${}_{a}$ with strongly singular potential ${a\left|x\right|}^{-2}$ on a bounded domain $\Omega $ are considered. If $\Omega ={\mathbb{R}}^{N}$ and $a>-{(N-2)}^{2}/4$, then the global existence of weak solutions is confirmed by applying the energy methods established by N. Okazawa, T. Suzuki, T. Yokota (2012). Here $a=-{(N-2)}^{2}/4$ is excluded because $D\left({P}_{a\left(N\right)}^{1/2}\right)$ is not equal to ${H}^{1}\left({\mathbb{R}}^{N}\right)$, where ${P}_{a\left(N\right)}:=-\Delta -{(N-2)}^{2}/{\left(4\right|x|}^{2})$ is nonnegative and selfadjoint in ${L}^{2}\left({\mathbb{R}}^{N}\right)$. On the other hand, if $\Omega $ is a smooth and bounded domain with $0\in \Omega $, the Hardy-Poincaré inequality is proved in J. L. Vazquez, E. Zuazua (2000)....